IntechOpen is the world's leading publisher of Open Access books, built by scientists for scientists. With 7,200 Open Access books available, 191,000 international authors and editors, and 210 million downloads, IntechOpen offers a platform for scientific research and knowledge sharing. The Fokker-Planck equation is a key concept in stochastic processes, describing the evolution of conditional probability density for a Markov process governed by an Itô stochastic differential equation. This chapter summarizes the derivation of the Fokker-Planck equation through various methods, including elementary proofs and those based on stochastic process theory. It also discusses the application of the Fokker-Planck equation in analyzing the Stochastic Duffing-van der Pol (SDvdP) system, a complex dynamical system. The chapter outlines the evolution of conditional probability density, the Kolmogorov forward and backward equations, and the use of the Fokker-Planck equation in stochastic control and prediction. The analysis includes numerical simulations of the mean and variance evolutions of the SDvdP system, demonstrating the effectiveness of different approximation methods. The chapter concludes by emphasizing the importance of the Fokker-Planck equation in understanding and analyzing stochastic systems across various fields.IntechOpen is the world's leading publisher of Open Access books, built by scientists for scientists. With 7,200 Open Access books available, 191,000 international authors and editors, and 210 million downloads, IntechOpen offers a platform for scientific research and knowledge sharing. The Fokker-Planck equation is a key concept in stochastic processes, describing the evolution of conditional probability density for a Markov process governed by an Itô stochastic differential equation. This chapter summarizes the derivation of the Fokker-Planck equation through various methods, including elementary proofs and those based on stochastic process theory. It also discusses the application of the Fokker-Planck equation in analyzing the Stochastic Duffing-van der Pol (SDvdP) system, a complex dynamical system. The chapter outlines the evolution of conditional probability density, the Kolmogorov forward and backward equations, and the use of the Fokker-Planck equation in stochastic control and prediction. The analysis includes numerical simulations of the mean and variance evolutions of the SDvdP system, demonstrating the effectiveness of different approximation methods. The chapter concludes by emphasizing the importance of the Fokker-Planck equation in understanding and analyzing stochastic systems across various fields.